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AMEEICAN SOCIETY OF CIVIL ENGINEERS. 

/ INCORPORATED 185 2. 



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DISCUSSION 

At the Seventh Annual Convention.* 

■.'kD — 



■ FLEXURE OF BEAMS. 



Mr. Egbert H. Thurston. — Eef erring to ''Eesistancesof Beams to 

M 

Flexure " : — f 

1°. I agree fully with Gen. Barnard in considering the formulse of 
Navier, and those in common use as based upon them, as well as the ar- 
guments of Decomble sustaining the former, as not well suiDported by 
the results of experiment, except in a few special cases. 

2^. The ordinary theory, and its resulting equations, in which the 
resistances of particles to compression and to extension are proportional 
to their distance from the neutral surface, are apparently sufficiently 
correct up to that limit of flexure at which the exterior sets of particles 
on the one side or on the other, are forced beyond the elastic limit. 

30, With absolutely non-ductile materials, or materials destitute of 
viscosity, fracture occurs at this point. But, with ordinary materials, 
and notably with good iron, low steel, and all of the useful metals and 
alloys in common employ, rupture does not then take place. 

40. The exterior portions of the mass are compressed on the one side, 
offering more and more resistance nearly, if not quite, up to the point of 
actual l:)reaking, which breaking may only occur long after passing the 
elastic Hmit. On the other side, the similar sets of particles are drawn 
apart, passing the elastic limit for tension, and then resisting the stress 

* Referring to— Record of Experiments shewing the Character and Position of Neutral 
Axes, as shown by polarized Light, L. Nickerson, Vol. Ill, page 31 ; and to Resistance of 
Beams to Flexure, J. G. Barnard, Vol. Ill, page 123. t Vol. Ill, page 123. 



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with approximately constant force, " flow " occurring until that limit of 
flow is reached, and rupture takes place. 

5°. Fracture may occur under either of several sets of conditions. 

A. The material may be absolutely brittle, {a. ) In this case, the 
elastic limit and the limit of rupture coincide for both simple tension and 
simj)le compression. The piece w^ill break with a snap when, under flex- 
ure, either limit is reached, (6. ) Or, it may happen that the limit is 
reached simultaneously on both sides. 

B. The material may be slightly viscous, {a.) The flexure of the piece 
will produce compression or extension, or both, beyond the elastic limit 
before rupture, giving three sets of conditions to be expressed by the 
formrda. (5.) The increase of resistance, after passing the elastic limit 
will not be similar for both forms of resistance, and each substance will 
probably be found characteristically distinguishable from every other. 
(c.) It would appear from exi^eriments already familiar, that the resist- 
ance to compression wiU frequently increase in a very high ratio as com- 
pared with that to extension, thus swinging the neutral surface toward 
the compressed side, and probably sometimes approximately to the limit- 
ing surface, with very hard and friable substances, thus bringing about 
something like a correspondence with " Galileo's theory." This, I j)re- 
sume, does not often happen. 

C. The material may be very ductile or viscous.* {a.) In this case 
the phenomena of flexure and rupture will be as last described, but of 
exaggerated extent and importance, {b. ) The resistances to extension 
and to compression as developed in this case, wiU be approximately, or 
accurately, those observed in experiments producing rupture by direct 
tension and by direct compression. The neutral surface will be deter- 
mined in position by the ratio of these ultimate resistances. 

6°. Proposition 1, of Decomble, as rendered by Gen. Barnardf, there- 
fore, may or may not be true for any individual case, and it cannot be 
true for all materials. Proposition 2 is, I think, probably correct, it be- 
ing understood that the effect of "flow" in producing modification of 
the co-efficients of elasticity and of rupture is comprehended. Proposi- 
tion 3, is, I should say, certainly incorrect for ductile substances. 

Ducomble is, therefore, in error in claiming that Navier's theories, 
narrow and inflexible as are their conditions, explain "all phenomena" 



* It is to be remembered that vigcosity and high cohesive force may co-exist, as shown by 
Prof. Henry and Mon. Presca. 
t Vol. Ill, page 123. 



of €exnre aud rupture, or that it can always give us correct moduli of 
rupture, or that it is in "complete harmony" with any but a narrow 
range of practice. 

7°. The statement that "any load, however small," is "capable of 
producing rupture i)roviding that the tiial is sufficiently x^rolonged," I 
have long since shown, by experiment (which has been published in 
this country and in Europe),^ to be quite the reverse of the truth in 
the case of iron, steel, etc. The fact, as shown by the fac- simile strain- 
diagTams illustrating these jjapers, being, that sialic stress, less than that 
producing rupture, hut greater than that corresponding with the elastic limit, 
produces actual increase of resisting power. This fact has since been 
proven by other investigators and by quite independent methods of 
research. 

8°. I have also shown in those experimental investigations, that the 
converse fact exists, that distortion, rapidly produced, causes an actual 
decrease of resisting power. Strain diagrams w^ere given illustrating this 
fact very strikingly. 

9°. This variation of resistance with variation of the method of rup- 
ture introduces another element of uncertainty into " Navier's theory," 
as well as into all formulas yet constructed. This element must remain 
until exj)eriment has indicated a measure of it and the form of the func- 
tion expressing its law, and thus enable us to construct a correct formula. 

10*^. Referring to the remarks of Gen. Barnard which follow the 
paper under discussion, f w^e may find in the phenomena just considered 
a reason for the fact, remarked by him, that "beams fractured by shot 
did not resist anything like so much " as those broken under the slow 
and steady action of the hydraulic press. 

11°. The assumption that resistances vary each way from the neutral 
surface proportionally with their distance from that surface, is when 
coupled with a rejected hypothesis of Navier, nevertheless, not far from 
the truth in special cases, as may be shown by proper mathematical 
treatment and comparison with results obtained experimentally. 

12°. Mr. William KentJ made this comparison for cast and wrought 
iron and for ash. The results of analysis and of experiment give the 
following values of the R in the ordinary formula: | 

* Transactions, Vol. II, page 239 ; Vol. Ill, page 12, &c. ; Journal of the Franklin Insti- 
tute, 1874 ; Van Nostrand's Engineering Magazine, 1874 ; London Engineering, 1873 ; 
Practical Mechanics' Magazine, ISli; Dingier' s Poly technisches Journal, 1875. 

t Vol. Ill, page 127. 

t. Member of Steven's Institute op Technology. § Wood on Resistance of Materials. 



M=iR B D^ 

for a beam fixed at one end, loaded at the other 



(1) 



R — (theoretical) 

R — (experimental) . 



CAST IBON. 



j \\T10UGHT IKON. 



32 280 

35 000 



60 000 
60 000 



12 120 
12 000 



13°. This remarkable approximation is thus derived 
fixed beam, Fig. 5, with loaded extremity, the force P 
being a measure of the weight W, and the beam having 
a depth D, a breadth unity, and a neutral surface sit- 
uated at a distance Y from the superior surface of the 
beam. Representing the resistances graphically by 
the triangles T JV, C 2^; their measures in tension and 
compression are respectively \ T N, \ G N, and their moments are 




\ T N Xi Y = \ T Y\ ij^ndi h G N X^i [D - Y) 



G{D— Y}"". 



An early hypothesis of Navier, which seems to have been entirely 
abandoned by him subsequently, and which has not been accepted 
by subsequent writers on the subject, make these moments equal. 
Assuming this to be correct, 

TY''=^G[D—Y}'' ^ . . • (2) 



G F2 _ 



(3) 



and, from this expression, we may find the position of the neutral sur- 
face, as determined by the assumed conditions. Then, letting B = the 
breadth of the beam, 

WL = iB [TY"- ^ G{D—Yf]=\RBD'' (4) 

in which latter expression R is the modulus of rupture, and its value 
can be found when G and T are known. It will always be of a value 
intermediate between T and G. 

140. The following are the data and results for the three cases taken: 
the results are well worthy of examination and record. 



Cast iron 

Wrought iron 
Ash timber . . , 



T 


c 


^ 


D 


L 


F 


D- Y 


WL 


16 000 


96 000 


1 


1 


1 


0.71 


0.29 


5 380 


60 000 


60 000 


1 


1 


1 


0.5 


0.5 


10 000 


17 200 


9 000 


1 


1 


1 


0.42 


0.58 


2 020 



32 280 
60 000 
12 120 



5 

15'^. The common theory of rupture, as it is defined by Prof, Wood, 
is confessedly far from correct, and, as shown at the beginning of these 
remarks, the neutral surface must vary in i)osition, and cannot invariably 
pass through the centre of gravity of section. It would seem that such 
coincidence of position, when occurring at all, is simply a matter of in- 
cidental concurrence of conditions. I therefore consider the criticism of 
Gen. Barnard to be just in this point. 

16^. The accurate mathematical expression of the phenomena of 
flexture and rupture, as already remarked, must be vastly more compre- 
hensive and flexible, and more facile of application than any yet proposed. 
As I have shown, it is not sufficient that both R^ and R,. — i. e., both T 
and C — appear in the formulas, as proposed by Decomble. The real 
value of these quantities, as there apjDearing, must vary as some function 
of distance from the neutral line, while the position of the neutral line 
must itself vai*y T\-ith both the value of Tand C in different cases, and 
with their change of value in the same beam, as flexure progresses and 
after rujDture commences. These variable functions must all be taken 
into account and comprised in the general expression for the moment 
resisting fracture. 

The characters of these functions, however, are nnfortunately not yet 
ascertained, and it is only after experiments in which the moment of re- 
sistance is accurately measured during every stage of fracture, and 
so completely that the strain diagram of the experiments can be 
graphically given, or its equation constructed, that we can obtain their 
values. This has, as yet, been done in but few^ cases, as in some experi- 
ments of Hodgkinson, in the work of Styffe, and in experiments made by 
Eodman. 

17°. It does not necessarily follow that the formulas finally resulting 
must be either complex or inconvenient of apx)lication. Simple expres- 
sions will, at least, be found for sj)ecial cases of simple character, which 
will serve every purpose of the engineer. 

18'^ It is also true, as remarked by Prof. Wood, and as known by 
every engineer, that the phenomena of flexure within admissible limits 
are much less comjDlex and much less difficult of manij)ulation than those 
of rupture, or than those resulting in serious permanent distortion. 
Hence, it is true that ordinary engineering i^ractice is not placed at such 
a serious disadvantage as these defects of the theory of strain might seem 
to indicate. 



19°. In common with every member of the profession, I am called 
upon to admit the great services rendered us by Navier in the splendid 
work done by him at VEcole cles Pants et Chaussees, in establishing a 
theory of engineering, as well as in working up a theory of rupture, and 
I desire to acknowledge those services, while declining to admit absolute 
accuracy in his theories. They were constructed at a time when science 
was apparently divorced from the practice of engineering, and when his 
services in securing a genuine union were most invaluable. He must 
always be regarded as one of the great leaders in our x^rofession. 

I would unite with Gen. Barnard in his remarks, relative to the at- 
tempt of Napier's pupil, Decomble, to retain the theory while modifying 
the formulas of Navier: — "If by discarding a coefficient founded upon 
an imaginary coefficient of elasticity, and the introduction of distinct 
and independent factors, symbolic of resistance to rupture by compres- 
sion and extension, it is shown that the Navier formula can be made 
reliable, an important service has been rendered to engineering 
science."^ 

I would myself add that the discovery and the mathematical expres- 
sion of the varying functions w^hich I have described, and the establish- 
ment of formulas of application embodying the facts of the variation of 
the coefficient of elasticity, of that of the module of resistance of rupture 
by tension and compression, and in the xDOsition of the neutral surface 
which are still as pre^dously essential, but unknown elements of a cor- 
rect theory of strain ; all of these yet remain to compensate some skill- 
ful experimenter and expert analyst. Their determination would earn 
for their fortunate discoverer higher distinction than ever won by either 
Coulomb or Navier. 



*Vol. Ill, page 126. 



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HoUinger Corp. 

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